Runoff on rooted trees

TL;DR

This paper presents a simplified model for overland runoff on hill-slopes, demonstrating a phase transition in runoff behavior driven by rainfall rate and stream coalescence, with implications for understanding flood dynamics.

Contribution

It introduces a novel idealized model linking runoff coalescence to phase changes in overland flow, highlighting the impact of coalescence on runoff thresholds.

Findings

Runoff increases dramatically when rainfall exceeds a critical threshold.

Stream coalescence lowers the critical rainfall rate for runoff onset.

Total runoff depends on the degree of stream coalescence.

Abstract

We introduce an idealised model for overland flow generated by rain falling on a hill-slope. Our prime motivation is to show how the coalescence of runoff streams promotes the total generation of runoff. We show that, for our model, as the rate of rainfall increases in relation to the soil infiltration rate, there is a distinct phase-change. For low rainfall (the subcritical case) only the bottom of the hill-slope contributes to the total overland runoff, while for high rainfall (the supercritical case) the whole slope contributes and the total runoff increases dramatically. We identify the critical point at which the phase-change occurs, and show how it depends on the degree of coalescence. When there is no stream coalescence the critical point occurs when the rainfall rate equals the average infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the average infiltration rate, and increasing the amount of coalescence increases the total expected runoff.